In this article, the stochastic Itô Volterra integral equation has been solved using long short-term memory (LSTM) based neural networks. The Monte Carlo (MC) based approach has been implemented to discretize the integral equations. The suggested method discretizes the set of stochastic integral equations into a system of linear algebraic equations. Numerical examples for different circumstances have been provided to illustrate the flexibility of our proposed solutions. An LSTM-based neural network model has been implemented with Adam optimization to obtain solutions. The efficiency of our proposed technique has also been examined with the help of loss function graphs and real and predicted function value graphs.
We aim to extend some fixed-point results to the setting of b-locally convex spaces. As an application, we establish the existence and uniqueness of solutions for a class of nonlinear tempered ψ-fractional differential equations on unbounded domains. A key advancement of our result is that it does not impose boundedness conditions at infinity on the nonlinearity, thereby generalizing several existing results.
We explore a hybrid model that combines a parabolic integro-differential equation with a parabolic hemivariational inequality, in infinite-dimensional spaces under a non-Lipschitz condition for the term source. We establish the existence and uniqueness of mild solutions. As an application, we consider a contact problem with normal compliance to illustrate the abstract results.
A linear time interpolation of temperature and heat flux is implemented within a parabolic (time-dependent) fundamental solution-based scheme to solve transient heat transfer problems using the subdomain boundary element method (BEM). This interpolation approach represents the main novelty of the work. Additionally, the numerical accuracy of the schemes is evaluated across a wide range of Fourier numbers, rather than using a fixed Fourier number value as is common in most studies on transient heat transfer. This broader comparison helps identify which scheme is best suited for different heat transfer rates. We compare three numerical schemes: the well-established elliptic (Laplace) scheme, the parabolic scheme with constant fluxes, and the parabolic scheme with linear interpolation of both temperature and fluxes. Results from three computational examples demonstrate that all three methods yield high numerical accuracy. However, the newly implemented parabolic scheme with linear time interpolation achieves the highest accuracy, particularly at high Fourier numbers.
We are concerned with the positive solutions for a class of generalized Laplacian fractional integral boundary value problem with a parameter. The existence, nonexistence, and multiplicity of positive solutions are derived in terms of different values of the parameter. Our approach relies on the Guo–Krasnosel’skii fixed-point theorem on cone. An example is also given to illustrate the main results.
We cover the gap in the analysis of explosion time between one-dimensional and multidimensional case for a rather general class of stochastic Volterra integro-differential equations. By mimicking the classical techniques from Murge and Pachpatte (1986) and generalizing them to the multidimensional case, we prove that the class of equations under consideration has almost surely infinite explosion time.
We study the nonlinear fractional Schr & ouml;dinger equation (-1)alpha u + V (x)u = f (x, u), u E H alpha (DEBn, DEB), where (-1)alpha (alpha E (0, 1)) stands for the fractional Laplacian of order alpha, x E DEBn, V EC(DEBn,DEB) may change sign and f is only locally defined near the origin with respect to u. Under some new weak and general sublinear assumptions on the nonlinearity f (x, u), we show that the above system has infinitely many solutions near the origin. Some examples are also given to illustrate our main theoretical result.
This work investigates the stability properties of a new class of quadratic Erd & eacute;lyi-Kober-type integral equations of fractional-order, which have not been addressed in the existing literature. For the first time, a rigorous Ulam-type stability analysis is presented for their quadratic fractional counterparts. Sufficient and necessary conditions are established for various types of Ulam-type stabilities, including Ulam-Hyers stability, Ulam-Hyers-Rassias stability, and sigma-semi-Ulam-Hyers stability. The analysis is based on Banach's fixed-point theorem combined with the Bielecki metric, which is well suited to handling nonlinearities and the nonlocal nature of fractional integrals. To support the theoretical findings, several illustrative examples are provided.
The Gurtin-Pipkin equation for heat conduction is a convolution type of integrodifferential equation which serves as a modification of the classical heat equation. It was proposed to overcome the unpleasant feature of infinite speed of propagation of the classical heat equation. It is well known that the classical heat equation is exponentially stable, and this extends to the abstract case where a negative definite linear self-adjoint operator takes place of the Laplacian. We study exponential stability of the Gurtin-Pipkin-type of abstract heat equation in the setting of Hilbert space. Sufficient conditions for exponential stability are established and this is done by using a semigroup approach. An illustrative example is given, and numerical simulations are also provided which are in good agreement with the theoretical analysis.
We investigate the existence of solutions for a class of semilinear integrodifferential evolution equations with infinite delay and infinite state-dependent delay in Banach spaces. We employ a new fixed-point theorem that relies on the degree of nondensifiability. To illustrate our findings, we present an illustrative example that demonstrates the key outcomes of our analysis.
We explore some aspects of two-index stochastic fractional differential equations (TSFDE). Under the global Lipschitz condition, we first investigate the existence and uniqueness of solutions, demonstrating that the solution depends continuously on the initial condition, the fractional orders, and time. Subsequently, we extend the existence and uniqueness results to cases governed by a local Lipschitz condition. Finally, we propose the theta-Euler-Maruyama (EM) scheme for TSFDE and establish its strong convergence at a certain rate in the Lp-norm under a global Lipschitz condition, as well as its strong convergence under a local Lipschitz condition.
This paper investigates the boundary behaviour of potential-type integrals for the multi-term time-fractional diffusion equation (MTFDE) across the moving boundary. First, we establish the jump relation for the integral operator associated with the fundamental solution of the inhomogeneous MTFDE. Second, we prove the continuity of the integral operator generated by the kernel corresponding to the homogeneous MTFDE. Krasnoschok obtained similar results for the time-fractional diffusion equation. However, in the multi-term case, the fundamental solution has more complex structure and does not admit standard scaling properties, which requires a different approach. Our results are essential for the analysis of boundary integral equations related to the MTFDE in time-dependent domains.
This study establishes the adequate conditions for the solvability of a Volterra-type nonlinear system of integral equations with some general kernels. The proof of the existence result is established in the space the methodologies for noncompactness measures are employed in our analysis. Moreover, we study the Cauchy problem for a system of fractional differential equations. Finally, we present some examples in sequence spaces that demonstrate the application of our results to specific kernels and a class of more general kernels that satisfy a specific condition.
The piecewise polynomial based Galerkin, multi-Galerkin methods and their iterated versions are applied to approximate the system of nonlinear second kind Fredholm-Urysohn integral equations and obtain superconvergence rates in both the case of smooth and weakly singular algebraic and logarithmic type kernels. We derive theorems in connections with convergence rates for Galerkin, multi-Galerkin methods and their iterated version in the uniform norm and show that the iterated versions provide better approximations as in the case of single Fredholm-Urysohn equation. Numerical examples are provided for justifying the reliability and efficiency of the theoretical results.
We study approximation of solutions to a class of first order retarded type functional differential equations with noninstantaneous impulses. The proposed problem is restricted to a finite dimensional subspace by using the projection operators. We establish a sufficient condition that guarantees the existence and uniqueness of approximate solutions. Our main results are developed by utilizing analytic semigroup theory, fixed point theorem, and Gronwall inequality. Furthermore, we study Faedo-Galerkin approximations and their convergence. Lastly, we provide a simulated example demonstrating the applications of the obtained theoretical results to partial differential equations.
We examine a viscoelastic Timoshenko system with infinite memory terms affecting the shear force. We demonstrate that the system remains stable for a wide class of relaxation functions and establish a relationship between the decay rate of the solutions and the growth of g at infinity. We derive a general decay result using the multiplier method with some additional arguments. Our findings expand upon and enhance several previous results in the literature.
We study the existence and uniqueness of solutions for a new fractional nonlinear integro-differential equation with variable coefficients and a functional boundary condition using Krasnoselskii's fixed point theorem and Banach's contractive principle. The approach relies on an inverse operator in a Banach space, the Mittag-Leffler function and an implicit integral equation. The technique used has many applications to studying various nonlinear integral or differential equations, including partial differential equations, with initial or boundary conditions. Several illustrative examples are provided to show applications of our main theorems by computing approximate values of the Mittag-Leffler functions.
The inverse black body radiation problem is mathematically the inverse problem of a Fredholm integral equation of the first kind defined on infinite interval, which is ill-posed. In this paper, a two-parameter regularization method with Tikhonov regularization and maximum entropy for the inverse black body problem is proposed. We prove that the regular solution obtained by the two-parameter regularization method converges to the true solution. Then, we give a numerical calculation scheme through discrete integration, and use generalized cross validation method to determine the two parameters. The numerical inversion results show that the proposed two-parameter regularization method is stable.
We study backward doubly stochastic differential equations driven by Teugels martingales associated with nonhomogeneous L & eacute;vy processes. We prove the existence and uniqueness of solutions by using the predictable representation theorem for nonhomogeneous L & eacute;vy processes and applying a Picard iteration method under stochastic Lipschitz conditions. Additionally, through a comparison principle, we establish the existence of a minimal solution under the assumption of continuous coefficients with stochastic linear growth.
